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For a graph $G$, a zero-sum $k$-flow is an assignment of labels from the set $\{\pm 1, \pm 2,\dots, \pm (k-1)\}$ to the edges of $G$ such that the total sum of all edges incident with any vertex of $G$ is zero.
The authors give a necessary and sufficient condition for the existence of a zero-sum flow for a given graph and they prove that every 2-edge connected bipartite graph has a zero-sum 6-flow, every cubic graph has a zero-sum 5-flow and every regular graph of even degree has a zero-sum 3-flow. The authors further conjecture that every graph which has a zero-sum flow, has a zero-sum 6-flow; this conjecture is implied by the Bouchet conjecture.
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